# Number 3

Three is approximately π when doing rapid engineering guesses or estimates. The same is true if one wants a rough-and-ready estimate of e, which is actually approximately 2.71828. Three is the first odd prime number, and the second smallest prime. It is both the first Fermat prime and the first Mersenne prime, the only number that is both, as well as the first lucky prime. However, it is the second Sophie Germain prime, the second Mersenne prime exponent, the second factorial prime, the second Lucas prime, the second Stern prime. Three is the first unique prime due to the properties of its reciprocal. Three is the aliquot sum of 4.

Three is the third Heegner number.

According to Pythagoras and the Pythagorean school, the number 3, which they called triad, is the noblest of all digits, as it is the only number to equal the sum of all the terms below it, and the only number whose sum with those below equals the product of them and itself. Three is the second triangular number and it is the only prime triangular number. Three is the only prime which is one less than a perfect square. Any other number which is n2 − 1 for some integer n is not prime, since it is . This is true for 3 as well, but in its case one of the factors is 1. Three non-collinear points determine a plane and a circle.

Three is the fourth Fibonacci number. In the Perrin sequence, however, 3 is both the zeroth and third Perrin numbers. Three is the fourth open meandric number.

Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions, A natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three and the sum of its digits is 2 + 1 3. Because of this, the reverse of any number that is divisible by three is also divisible by three. For instance, 1368 and its reverse 8631 are both divisible by three . See also Divisibility rule. This works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one . A triangle is the only figure which, if all endpoints have hinges, will never change its shape unless the sides themselves are bent. 3 is the smallest prime of a Mersenne prime power tower 3, 7, 127, 170141183460469231731687303715884105727. It is not known whether any more of the terms are prime. Three of the five regular polyhedra have triangular faces — the tetrahedron, the octahedron, and the icosahedron. Also, three of the five regular polyhedra have vertices where three faces meet — the tetrahedron, the hexahedron, and the dodecahedron. Furthermore, only three different types of polygons comprise the faces of the five regular polyhedra — the triangle, the quadrilateral, and the pentagon. There are only three distinct 4×4 panmagic squares.

Only three tetrahedral numbers are also perfect squares.

The first number, according to the Pythagoreans, and the first male number. The first number, according to Proclus, being the first number such that n2 is greater than 2n. The trisection of the angle was one of the three famous problems of antiquity. 3 is the second triangular number.

Gauss proved that every integer is the sum of at most 3 triangular numbers. Gauss proved that for any prime number p the product of its primitive roots is ≡ 1 . Any number not in the form of 4n is the sum of 3 squares.

In numeral systems

It is frequently noted by historians of numbers that early counting systems often relied on the three-patterned concept of "One- Two- Many" to describe counting limits. In other words, in their own language equivalent way, early peoples had a word to describe the quantities of one and two, but any quantity beyond this point was simply denoted as "Many". As an extension to this insight, it can also be noted that early counting systems appear to have had...

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